Homework #4 Stat. Mech. CHAP E4120x Prof. Ben O'Shaughnessy Homework #4 (a) A 3-D liquid crystalline fluid consists of N rod-like molecules of length...

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Homework #4 Stat. Mech. CHAP E4120xProf. Ben O’Shaughnessy1Homework #4(a) A 3-D liquid crystalline fluid consists of N rod-like molecules of length l.Each rod can occupy one of V available lattice sites, i.e., the center of gravity ofeach rod can be at one of the V lattice sites indicated by the dots in the first figure.The rods can orient themselves arbitrarily. The number of directions each rodcan point in per unit solid angle is l. Consider a dilute system (N(b) The system described in part (a) is brought into contact with a zeolite (aporous material). The particular zeolite consists of many cylindrical tubesand each tube contains a 1-D array of L lattice sites lying on its center axis (asshown in the second figure).dSystem (a):N rod moleculesV lattice sitesL sites per tubeVz sites in the zeolite V sites in the solutionHomework #4 Stat. Mech. CHAP E4120xProf. Ben O’Shaughnessy2The total number of lattice sites in all the tubes in the zeolite is Vz. The rodsare free to move between the two systems so one rod can now occupy any of thelattice sites both inside and outside the zeolite. The interesting problem ariseswhen the diameter of the tubes, d, is smaller than l, so the rods in the tubes havefewer ways to orient themselves because of the constraint of the walls of thetubes. Calculate the number of rods, M, absorbed by the zeolite in equilibrium.Hence determine a relation between the densities in the zeolite, φz , and in thesolution outside the zeolite, φs , where the density in each region is defined as thenumber of rods per site in that region.Hint for the number of directions in a 3-D space: A rod with indistinguishableends which can orient freely within an angle f of a certain direction (indicated bythe cones around the z axis in the figure below) has W possible orientationalconfigurations, where Ω = 2πλ (1− cosφ ) .Hint for (b): Find the number of directions a rod can point in when it isabsorbed by the zeolite. Express the entropy of the whole system as a functionof the number of rods absorbed by the zeolite and then find M by maximizing thisentropy.fz